Proof that \( \sqrt{3} - \sqrt{5} \) is an Irrational Number
- Assume the Contrary
Let's assume that \( \sqrt{3} - \sqrt{5} \) is a rational number. This means it can be expressed as a fraction in the form \( \frac{a}{b} \), where \( a \) and \( b \) are integers with no common factors.
- Square both Sides
Squaring both sides of the equation \( \sqrt{3} - \sqrt{5} = \frac{a}{b} \), we get:
\( 3 - 2\sqrt{15} + 5 = \frac{a^2}{b^2} \)
\( 8 - 2\sqrt{15} = \frac{a^2}{b^2} \)
- Isolate the Irrational Part
Now, isolate the irrational part on one side of the equation:
\( \sqrt{15} = \frac{8b^2 - a^2}{2b^2} \)
- Contradiction
This contradicts the fact that \( \sqrt{15} \) is irrational, as it cannot be expressed as a ratio of two integers. Therefore, our assumption that \( \sqrt{3} - \sqrt{5} \) is rational is false.
- Conclusion
Hence, we have proven that \( \sqrt{3} - \sqrt{5} \) is an irrational number.